The \({L}_{2,\infty }\) -estimate is deduced from the estimates in the \(p>2\)  case and \(p<2\)  case by an interpolation argument. The proof of the estimates in the \(p<2\)  case essentially follows the approach of [1]} to the Cwikel-type estimates on NC Euclidean spaces. As it turns out, on noncommutative tori some simplifications occur, which allows us to get much sharper results. In fact, as far as Schatten and weak Schatten classes are concerned, the estimates of this papers are as sharp as in the (commutative) Euclidean space setting.
